Critical dynamics in complex systems are characterized by power-law distributions of spatial and temporal properties of system events. The occurrence of critical dynamics in neuronal networks is increasingly supported by multielectrode array recordings of spontaneous activity in organotypic cortical slice cultures. System events in neuronal networks are typically defined as activations of neuronal ensembles, or "neuronal avalanches". Although studies associate critical neuronal network avalanche dynamics with optimized information transfer, the directed nature of information transfer in these networks has not been previously examined. Here we present three novel transfer-entropy based measures of directed information transfer in neuronal avalanches. Our measures compute the amount of predictive information present in avalanche properties of the source region about avalanche properties of the destination region and are suitable for detecting information transfer at multiple spatial scales, from individual neurons to neuronal ensembles. We apply these measures to compute directed information transfer in large, sparse, modular networks of leaky integrate-and-fire neurons with spike timing-dependent synaptic plasticity and axonal conduction delays. We find that all three measures peak at criticality in all examined networks. Our findings pave the way for the application of our measures to empirical multielectrode recording data.